The numbers of prison reform depend on how you count

Most people currently in prison have violent records — even though most people who are sent there do not.

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In their conventions over the last two weeks, the Democratic and Republican parties offered starkly contrasting takes on the state of criminal justice. The Democratic platform set goals of ending the “War on Drugs” and the “era of mass incarceration.” Republicans, standing pat on their 2016 platform, pledged “gratitude and support” to law enforcement and promised an end to “politicized second-guessing.”

As the election approaches and these visions collide, expect to hear two seemingly contradictory statistics from politicians and their supporters.

One: In 2019, only 5 percent of arrests in the United States — and only about 25 percent of prison admissions — were for violent offenses, with nonviolent drug crimes representing by far the largest share both of arrests and sentences.


Two: The majority of people in prisons, about 53 percent, are there because of violent criminal convictions.

So, which is it? Are most incarcerated people convicted of nonviolent offenses, or are most people in prisons serving sentences for violent crimes?

Actually, both are true. The apparent incompatibility of these numbers is an example of what’s called “stock and flow” confusion. The people sentenced to prison are the “flow,” representing the change to the population over time; the people currently in prison are the “stock,” a snapshot of the population at a given moment. The types of convictions represented in those categories won’t necessarily be identical because people convicted of violent crimes tend to serve longer sentences.

To understand how this could skew the stats, imagine a hotel with four rooms: two occupied by long-term residents who are staying for a year, and the others having a different guest each night. On any given night, the guests are equally split between long-term residents and nightly visitors. Over the year, though, 732 people would have come through the hotel, of whom only two were the long-termers. Stock-and-flow confusion can occur whenever two subgroups or entities have different lengths of time between entering and leaving. The food in your pantry is mostly canned even though most of what you buy at the grocery store is perishable.


When it comes to prison reform, both stock and flow are important. The United States has a unique problem of mass incarceration measured both by how many people are currently in prison and by how many people cycle through the system each year and are thereafter branded by it. Both dynamics are made worse for Black Americans because of the systemic racial bias affecting arrest rates and duration of sentences. Ending the War on Drugs won’t make a sizable dent in the stock but will dramatically affect the flow, and improve millions of lives in the process. The fact that most people in prison at any given time are serving long sentences, though, isn’t evidence that violent crime is common or that the system works properly. It could just be math.

Aubrey Clayton is a mathematician living in Boston and the author of the forthcoming book “Bernoulli’s Fallacy.” Follow him on Twitter @aubreyclayton.